Showing papers on "Piecewise published in 1970"

Elements of Large-Scale Mathematical Programming Part I: Concepts

Abstract: A framework of concepts is developed which helps to unify a substantial portion of the literature on large-scale mathematical programming. These concepts fall into two categories. The first category consists of problem manipulations that can be used to derive what are often referred to as “master” problems; the principal manipulations discussed are Projection, Inner Linearization, and Outer Linearization. The second category consists of solution strategies that can be used to solve the master problems, often with the result that “subproblems” arise which can then be solved by specialized algorithms. The Piecewise, Restriction, and Relaxation strategies are the principal ones discussed. Numerous algorithms found in the literature are classified according to the manipulation/strategy pattern they can be viewed as using, and the usefulness of the framework is demonstrated by using it see Part II of this paper to rederive a representative selection of algorithms. The material presented is listed in the following order: The first section is introductory in nature, and discusses types of large-scale problems, the scope of discussion and the literature, and the notation used. The second section, entitled “Problem Manipulations: Source of 'Master' Problems” covers the subjects of projection, inner linearization and outer linearization. The third section, “Solution Strategies: Source of 'Subproblems',” discusses piecewise strategy, restriction and relaxation. The fourth section is entitled “Synthesizing Known Algorithms from Manipulations and Strategies,” and is followed by a concluding section and an extensive bibliography.

Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions

Abstract: This paper is concerned with the existence of weak solutions to certain nonlinear hyperbolic Cauchy problems. A condition on the curves of discontinuity is used which guarantees uniqueness in the class of piecewise smooth weak solutions. The method of proof is geometric in nature and is constructive in the manner of A. Doughs and Wu Cho-Chun; that is, for certain types of initial data the method of characteristics is employed to construct piecewise smooth weak solutions. A limiting process is then used to obtain existence for bounded, measurable initial data. The solutions in some cases exhibit interesting, new phenomena. For example, a certain class of initial data having one jump gives rise to a solution having a curving contact discontinuity which does not enter the region of intersecting characteristics. 1. Introduction. In this paper we consider existence of global weak solutions to the Cauchy problem

Diakoptics and Piecewise Methods

Abstract: Limitations on both high-speed core available and computer running time required for solving large-scale power system problems are very serious restrictions that continually confront the power system analyst. Piecewise methods, or methods of tearing as they are also called, hold much promise for the solution of such large scale problems. In this paper, new piecewise methods and algorithms are derived. Variations of these algorithms for practical applications are developed. Both boundary iteration as well as diakoptic methods are described. The major emphasis of this paper is placed on the latter method. The use of elimination and triangular factorization techniques within the piecewise methods is also indicated.

The optimization of trajectories of linear functional differential equations

Abstract: Our aim in this paper is to examine a number of fundamental questions in the theory of optimal control of processes monitored by certain general systems of linear functional differential equations with finite memories. In our model the controls may appear in a very general nonlinear functional manner which permits us to consider retardations of a rather general character in the control variables. In particular, we prove a maximal principle for such systems. We consider existence questions in the class of admissible Borel measurable (respectively piecewise continuous, almost piecewise continuous) initial functions and controls. We also show that certain solutions of an uncontrolled linear functional differential equation are piecewise analytic or quasi-piecewise analytic.

Constructing Wavefunctions for Nonlocal Potentials

Abstract: A method is developed for constructing wavefunctions for nonlocal interaction potentials of the type that arise in the theory of chemical reactions, electron scattering by molecules, and also in nuclear scattering problems The nonlocal potential is approximated by piecewise polynomial functions, for which analytic solutions are constructed Accurate and efficient algorithms are derived for evaluating all the special functions needed for the solutions The method is particularly advantageous for heavy‐particle scattering at higher energies

A consistent finite element model for the two-dimensional continuum

Abstract: The finite element approximation to the continuum problem is examined from the viewpoint of the principle of virtual work. It is shown that the usual nodal equilibrium equations for triangular elements are a consistent consequence of a piecewise constant strain field, thus guaranteeing that many results of general continuum theory can be directly applied to the finite element model, and also clarifying the relation between the two models.

The Dual Boundary Element Method ForDynamic Analysis Of Cracked Pin-loaded Lugs

Abstract: The dual boundary element method (DBEM) and the time domain method are applied for the determination of dynamic stress intensity factors (DSIF) for a general mixed-mode crack problem. The DBEM generates a distinct set of boundary integral equations by applying the displacement equation to one of the crack surfaces and the traction equation to the other. The present method does not require any domain discretisation. The boundary of the body is divided into quadratic elements and quarter-point elements (QPE) are used near the crack tips. The temporal variations of the boundary quantities are either piecewise constant or piecewise linear. The DSIF are calculated using the crack opening displacements of the QPE and a least-square error minimization. The method is applied to two problems. The solution for the first example is compared with the solutions obtained by using other methods and shows good agreement. The second example, a dynamic analysis of a pin-loaded cracked lug is an application of the method to a new mixed-mode problem.

Note on the design of linear systems with piecewise constant feedback gains

Abstract: The purpose of this correspondence is to report some relations that can be used to extend the scope of a computational algorithm for the suboptimal design of linear regulator systems. This algorithm was presented in [1], and can be used to determine suboptimal piecewise constant feedback gains to be used instead of the optimal continuously time-varying ones [3]. This correspondence deals with the extension to the case where, in addition to the piecewise constant gains, the switching times are also optimized.

Letter to the Editor-An Approach to Reducing the Computing Time for Dynamic Programming

Abstract: This note presents a procedure for reducing the computing time for dynamic programming when the optimal control function can be assumed to be piecewise smooth.

The bending of space curves into piecewise helical curves

Abstract: Regular C cubed space curve bending in Euclidean 3-space into piecewise helix, considering torsion and curvature

Controllability of linear time-invariant sampled-data systems with nonconstant shaping functions

Abstract: This paper is concerned with the controllability of single-input linear time-invariant plants after introduction of sampling in such a way that, for the time kT , the input is u_{k}f(t - kT) , where u k is a real constant and f(t) is a bounded piecewise continuous shaping-function given in 0 . Necessary and sufficient conditions for complete controllability of such sampled-data systems in terms of T and f(t) are given. This is a generalization of a well-known theorem for the special case f(t) \equiv 1 due to Kalman et al. The results may prove to be important if for practical reasons the shaping function f(t) \equiv 1 cannot be realized.

Computation of time-optimal controls using a second-variation descent search

Abstract: A second-variation descent search is proposed for computing time-optimal control functions for linear time-invariant plants. Since the piecewise constant structure of the optimal control function parameterizes the relevant two-point boundary value problem, the terms required in the variational analysis are simply first and second partial derivatives of an appropriate scalar cost function with respect to the parameters. Closed form expressions for these derivatives can be obtained. As a result, the second-order terms are incorporated into the analysis with only a modest increase in computational effort over that required for the first-variation methods appearing in the literature [1]-[4]. The usual second-variation algorithm can be applied only if the matrix of second derivatives is positive definite at each iteration. Since this condition generally fails to hold during the early stages of the search, a modified version of the algorithm is developed, based on the selection of a positive definite model for the matrix of second derivatives. Performance of the modified algorithm is shown to be significantly better than that of the familiar Newton algorithm in regions of parameter space far from the solution point. Particularly striking results are obtained when all the eigenvalues of the plant are real.

Piecewise solutions of electrical networks with coupling elements

Abstract: The piecewise method of network analysis developed by Kron under the name of “diakoptics” is adapted to account for coupling interactions between torn subnetworks using two different characterizations of electrical networks with coupling elements: (a) the standard network representation in which couplings between branches do not affect the topological description of the network but are included in the constitutive volt-ampere branch relations, appearing as off-diagonal terms in the branch immittance matrices; (b) the augmented network representation in which coupling elements appear as extra controlled-source branches acting as topological constraints and setting up the branch immittance matrices in diagonal forms. It is shown that the tearing algorithms of network solutions are remarkably more efficient when applied to augmented networks and, in particular, lead to simple computations of network sensitivities with respect to coupling parameters. Simple examples are presented to illustrate the basic features of various tearing algorithms.

Piecewise Linear Approximation of Imbeddings of Manifolds in Codimensions Greater than Two; Addendum to the Paper "piecewise Linear Approximation of Imbeddings of Cells and Spheres in Codimensions Greater than Two"

Abstract: It is shown that an arbitrary locally flat imbedding of one piecewise linear manifold into another of codimension greater than two can be changed to a piecewise linear imbedding by means of an arbitrarily small isotopy of the ambient manifold. Bibligraphy: 5 entries.

Piecewise Approximation of Functions of Two Variables and Its Application in Topographical Data Reduction.

Abstract: : A method to obtain variable-boundary piecewise polynomial approximations of functions of two variables f(x,y) is presented. A two state scan is used. In the first stage one uses the one-dimensional algorithm for a set of parallel directions in the domain of f(x,y). Then a similar algorithm is applied on the coefficients of the polynomials, i.e., it finds segments where groups of certain coefficients do not differ significantly from their average value. Computer tests of the algorithm on map data gave significant data reduction ratios (about 40:1) for mean square error of approximation around 2% of the relative altitudes. (Author)

Response of Viscoelastic Materials to Piecewise Continuous Excitations

Abstract: A general method is presented for predicting the response of a linear viscoelastic material to piecewise continuous excitations in terms of its known response to the standard excitation functions, i.e., in terms of the relaxation modulus and the creep compliance in case of non‐periodic excitations, or the real and imaginary parts of the complex modulus and complex compliance in case of the steady‐state response to a periodic non‐sinusoidal excitation. To illustrate typical behavior, the response is used of a three‐parameter Maxwell model to staircase, pyramid, triangular pulse, and triangular pulse train excitations.

Calculation of piecewise-constant control strategies for linear time-invariant systems

Abstract: A method is derived for computing piecewise-constant control strategies to transfer the state of a controllable linear system described by the vector-matrix equation dx/dt = Ax+Bu from the initial state x(0) to a specified state x(tn) in a specified time tn.

Stability analysis of a feedback system containing periodically varying parameters

Abstract: Stability analysis for a third-order feedback system with a periodically varying parameter is presented using a matrix method and approximating the time-varying parameter as piecewise constant over its period. The stability boundaries of such a system are predicted and compared with the boundaries determined by analogue computer simulation. Incidentally a method of generating similar matrices is brought out during this investigation.